Co-Ordinate Geometry Maths Studies

Distance Learning Objectives To find the distance of a line segment using two co-ordinates. To find coordinates when given 1 coordinate and the distance value. Determine the shape of a shape e.g. triangle using the distance formula.

Distance Example 1: Use the Distance Formula to find the distance between the points with coordinates (−3, 4) and (5, 2).

Distance √(-1- -2)²+(q – 4)² = √10 Example Find q given that P(-2,4) and Q(-1,q) are √10 units apart. √(-1- -2)²+(q – 4)² = √10 √(-1)² + (q – 4)² = √10 Square both sides to remove √ (-1)² + (q – 4)² = 10 1 + (q-4)² = 10 (q-4)² = 10 – 1 (q – 4)² = 9 (q – 4)² = √9 To remove the √ find the √ of the other side q – 4 = 3 q = 7

Midpoint Learning Objective To find the midpoint of a line segment using two co-ordinates. To find coordinates when given 1 coordinate and the midpoint.

Midpoint Find the midpoint R of the line segment with coordinates (-9,-1) and (-3,7)

Midpoint

Gradient The gradient of a line is a measurement of the steepness and direction of a nonvertical line.

Gradient Learning Objectives To find the gradient of a line given two coordinates. To find coordinates when given 1 coordinate and the gradient. Understand the relationship of parallel and perpendicular lines. Find the gradient of parallel and perpendicular lines

Gradient Formula Example M = Y2 – Y1 X2 – X1 A(1,5) B(2,8) 2 -1

Drawing a Line Draw a line that passes through the point A(1,4) with gradient -1 (see pg. 91 eg 2)

Gradient Perpendicular Lines Perpendicular lines have opposite gradients. Eg Gradient of AB = 2/3 Gradient of CD = -3/2

Gradient Parallel Lines Parallel lines have the SAME gradient e.g. Gradient of MN = 3/9 Gradient of OP = 1/3

Lines To prove that two lines are perpendicular, simply multiply their gradients. If the answer is -1, then the lines are perpendicular. Eg. The gradient of line AB is 2/3 The gradient of line CD is -3/2 2 x - 3 3 2

Equation of a line An equation of a line can come in different forms depending on their gradient, if they are horizontal or vertical and if they intercept the y-axis.

Equation of a line Example 1 – Gradient Intercept Form y = mx + c Where m = gradient and c is the y-intercept y = mx + c (0,c) intercept See Example 5 P. 95

Equation of a Line Example 2 – General Form ax + by + c = 0 To find the equation of a line in this form you need a gradient and one point. However, to find a gradient you need two points.

Equation of a line E.g. Find the equation of a line that joins the points A(-3,5) B(1,2) The gradient is m = 2 – 5 = - 3 1-(-3) 4 Using this formula y – y1=m(x – x1) y - -3 = -3/4(x – 5) 4(y + 3) = -3(x – 5) 4y + 12 = -3x + 15 3x + 4y – 3 = 0

Equation of a Line y - -3 = -3/4(x – 5) 4(y + 3) = -3(x – 5) Ans = 3x + 4y – 3 = 0